4 Nonautonomous Attractors
59
dist R d (φ (n, n 0 , D), A n ) → 0 as n → ∞ (n 0 fixed)
(4.4)
and a pullback attractor if it pullback attracts bounded subsets D of R
d , i.e.,
dist R d (φ (n, n 0 , D), A n ) → 0 as n 0 → −∞ (n fixed).
(4.5)
Random attractors [6] of random dynamical systems and snapshot attractors in
physics [26] are essentially pullback attractors.
The existence of a pullback attractor follows from that of a pullback absorbing
family.
Definition 4.4 A family B = {B n : n ∈ Z} of nonempty compact subsets of R
d is
called pullback absorbing if for every bounded subset D of R
d and n ∈ Z there exists
an N (n, D) ∈ N such that
φ (n, n 0 , D) ⊆ B n for all n 0 ≤ n − N (n, D).
It is said to be φ-positively invariant if φ
n, n 0 , B n 0
⊆ B n for all (n, n 0 ) ∈ Z
2
≥ .
The following theorem generalises of a well-known theorem on the existence of an
autonomous global attractor. The assumption about a φ-positively invariant pullback
absorbing family is not a serious restriction, since one can always be constructed
given a general pullback absorbing family [19].
Theorem 4.1 Suppose that a process φ has a φ-positively invariant pullback absorbing family B = {B n : n ∈ Z}.
Then there exists a global pullback attractor A = {A n : n ∈ Z} with component
sets determined by
A n =
j≥0
φ
n, n − j, B n− j
for all n ∈ Z.
(4.6)
Moreover, if A is uniformly bounded, then it is unique.
4.4 Construction of Forward Attractors
The situation is somewhat more complicated for forward attractors than for pullback
attractors due to some peculiarities of forward attractors [25], e.g., they need not be
unique.
Example 4.1 For each r ≥ 0 the process generated by
x n+1 = f n (x n ) :=
x n ,
n ≤ 0,
1
2
x n ,
n > 0
(4.7)
59
dist R d (φ (n, n 0 , D), A n ) → 0 as n → ∞ (n 0 fixed)
(4.4)
and a pullback attractor if it pullback attracts bounded subsets D of R
d , i.e.,
dist R d (φ (n, n 0 , D), A n ) → 0 as n 0 → −∞ (n fixed).
(4.5)
Random attractors [6] of random dynamical systems and snapshot attractors in
physics [26] are essentially pullback attractors.
The existence of a pullback attractor follows from that of a pullback absorbing
family.
Definition 4.4 A family B = {B n : n ∈ Z} of nonempty compact subsets of R
d is
called pullback absorbing if for every bounded subset D of R
d and n ∈ Z there exists
an N (n, D) ∈ N such that
φ (n, n 0 , D) ⊆ B n for all n 0 ≤ n − N (n, D).
It is said to be φ-positively invariant if φ
n, n 0 , B n 0
⊆ B n for all (n, n 0 ) ∈ Z
2
≥ .
The following theorem generalises of a well-known theorem on the existence of an
autonomous global attractor. The assumption about a φ-positively invariant pullback
absorbing family is not a serious restriction, since one can always be constructed
given a general pullback absorbing family [19].
Theorem 4.1 Suppose that a process φ has a φ-positively invariant pullback absorbing family B = {B n : n ∈ Z}.
Then there exists a global pullback attractor A = {A n : n ∈ Z} with component
sets determined by
A n =
j≥0
φ
n, n − j, B n− j
for all n ∈ Z.
(4.6)
Moreover, if A is uniformly bounded, then it is unique.
4.4 Construction of Forward Attractors
The situation is somewhat more complicated for forward attractors than for pullback
attractors due to some peculiarities of forward attractors [25], e.g., they need not be
unique.
Example 4.1 For each r ≥ 0 the process generated by
x n+1 = f n (x n ) :=
x n ,
n ≤ 0,
1
2
x n ,
n > 0
(4.7)